A polygon is any closed figure with sides made from straight lines. At each vertex of a polygon, there is both an interior and exterior angle, corresponding to the angles on the inside and outside of the closed figure. Understanding the relationships that govern these angles is useful in various geometrical problems. In particular, it is helpful to know how to calculate the sum of interior angles in a polygon. This can be done using a simple formula.

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Steps

1

Count the number of sides your polygon has. The method for calculating the sum of interior angles is based on how many sides the polygon has. Remember that a polygon must have at least 3 sides (a triangle), and each side must be a straight line.

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2

Subtract 2 from the number of sides. For example, subtracting 2 from a triangle gives you the number 1. Subtracting 2 from a pentagon (which has 5 sides) gives you the number 3. Subtracting 2 from a hexagon (which has 6 sides) gives you the number 4.

3

Multiply this number by 180. Multiply the number arrived at in the previous step by 180. This will yield the sum of the polygon's interior angles, expressed in degrees. For example, consider a hexagon. Subtracting 2 from a hexagon's 6 sides yields 4. Multiplying 4 by 180 yields 720. Therefore, a hexagon (regular or irregular) has interior angles that add up to 720 degrees.

4

Review the formula used to calculate this sum. Building a formula from the steps above yields: s = 180(n - 2), where "s" is the sum of the interior angles and "n" is the polygon's number of sides. This formula can be used for a polygon with any number of sides. It does not matter whether the polygon is regular or irregular, nor does it matter what the individual interior angles measure. Given a certain number of sides, a polygon's interior angles can always be summed using the formula above.

5

Derive the formula for further understanding. If you forget the formula above, or simply want to understand why it works, you can derive it fairly easily. This is done by segmenting any polygon into triangles.

Remember that a triangle's 3 interior angles will add up to 180 degrees, no matter what the shape of the triangle. This is the premise upon which the formula above can be built.

Draw a polygon on a sheet of paper. For example, consider a square, which is a 4-sided polygon. You can split the square up into 2 triangles by drawing a line connecting 2 of its opposite corners. Note that the interior angles of each triangle can be added to equal the interior angles of the original polygon.

Once you have split your polygon up into separate triangles, count the number of triangles in the polygon. Multiplying this number by 180 will yield the sum of the polygon's interior angles. The formula presented above makes use of this method.

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